Mortmere has alerted me that Star Trek's chairs and tables were a variation on the Tulip Chairs (below, left) and tables designed by Finland-born Eero Saarinen for Knoll in 1957. *
I went looking and sure enough, there they are.
Screencap from Star Trek episode "Is There in Truth No Beauty?" (1968).
*Mortmere also sent me the link to this web article "Chairs by Burke", which reports:
"The standard equipment chair seen on the sets of the Enterprise were manufactured by a company in Texas called "Burke". They were closely patterned after a similar chair by Saarinen except that they featured a four-blade 'propellor' base instead of the circular 'Tulip' base..."
You can see the base on the chair behind Spock's in this still from the 1966 episode "Charlie X." (It's harder than you might think to find a clear shot of an entire chair.) Eames' chairs also have a propellor base.
As always, Star Trek screencaps from TrekCore.com.